Paper:

Bounds on Pairs of Families with Restricted Intersections

We study pairs of families ${\cal A},{\cal B}\subseteq2^{\{1,\ldots,r\}}$ such that $|A\cap B|\in L$ for any$A\in{\cal A}$, $B\in{\cal B}$. We are interested in the maximalproduct $|{\cal A}|\cdot|{\cal B}|$, given $r$ and $L$. We giveasymptotically optimal bounds for $L$ containing only elementsof $s<q$ residue classes modulo $q$, where $q$ is arbitrary(even non-prime) and $s$ is a constant. As a consequence, weobtain a version of Frankl-R\"{o}dl result about forbiddenintersections for the case of two forbidden intersections. Wealso give tight bounds for $L=\{0,\ldots,k\}$.